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Brezis–Gallouet–Wainger remainder constant on the 2D torus (C16C_{16})

Brezis–Gallouet–Wainger remainder constant on the 2D torus: determine C16C_{16} or improve its bounds; the repository records β+ππ≈1.82283\frac{\beta + \pi}{\pi} \approx 1.82283 (lower) and ≈2.15627\approx 2.15627 (upper).

MSC 46E35 · Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

C16=LC_{16} = L is the smallest constant for which the sharp Brezis–Gallouet inequality

∥u∥L∞(T2)2≤14π ∥∇u∥L2(T2)2[ln⁡δ(u)+ln⁡(1+ln⁡δ(u))+L]\|u\|_{L^\infty(\mathbb T^2)}^2 \le \frac{1}{4\pi}\,\|\nabla u\|_{L^2(\mathbb T^2)}^2 \Bigl[\ln\delta(u) + \ln\bigl(1+\ln\delta(u)\bigr) + L\Bigr]

holds for all zero-mean functions u∈H2(T2)u \in H^2(\mathbb T^2) with sufficiently large frequency ratio

δ(u):=∥Δu∥L2(T2)2∥∇u∥L2(T2)2.\delta(u) := \frac{\|\Delta u\|_{L^2(\mathbb T^2)}^2}{\|\nabla u\|_{L^2(\mathbb T^2)}^2}.

Equivalently, LL is defined via the constrained extremal problem

L=max⁡δ≥1[4πΘ(δ)−ln⁡δ−ln⁡(1+ln⁡δ)]L = \max_{\delta \geq 1} \left[ 4\pi\Theta(\delta) - \ln\delta - \ln(1+\ln\delta) \right]

where Θ(δ):=sup⁡{∣u(0)∣2:∥∇u∥22=1, ∥Δu∥22=δ}\Theta(\delta) := \sup\lbrace|u(0)|^2 : \lVert\nabla u\rVert_2^2 = 1,\, \lVert\Delta u\rVert_2^2 = \delta\rbrace.

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Background

  • The leading coefficient 14π\frac{1}{4\pi} in front of the logarithmic terms is optimal, as is the doubly logarithmic correction; the remaining optimization is entirely in the additive constant LL.
  • The simpler "one-log" Brezis–Gallouet inequality ∥u∥∞2≤C∥∇u∥22(ln⁡δ+K)\lVert u\rVert_{\infty}^2 \le C\lVert \nabla u\rVert_{2}^2(\ln\delta + K) has infimum C=14πC = \frac{1}{4\pi}, but this infimum is not attained with any finite KK—the log-log correction is necessary.
  • The constant LL is expressed in terms of lattice sums over Z2\mathbb{Z}^2 and does not have a known closed form.
  • The maximum in the variational definition is unique and achieved at finite δ∗\delta^{\ast}; the corresponding conditional extremal uμ(δ∗)(x)u_{\mu(\delta^{\ast})}(x) is an exact extremal function.
  • Applications include sharp attractor dimension bounds for 2D Navier–Stokes equations on the torus.

References (from the repository page)

  • [BG1980] Brezis, H.; Gallouet, T. Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4 (1980), 677–681.
  • [BDZ2013] Bartuccelli, M. V.; Deane, J. H. B.; Zelik, S. Asymptotic expansions and extremals for the critical Sobolev and Gagliardo–Nirenberg inequalities on a torus. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 445–482. arXiv:1012.2061

For related results in Hölder space settings, see:

  • [MSW2010] Morii, K.; Sato, T.; Wadade, H. Brézis–Gallouët–Wainger type inequality with a double logarithmic term in the Hölder space: Its sharp constants and extremal functions. Nonlinear Anal. 73 (2010), 1747–1766.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound β+ππ≈1.82283\frac{\beta + \pi}{\pi} \approx 1.82283; best known upper bound ≈2.15627\approx 2.15627. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
≈2.15627\approx 2.15627 [BDZ2013] Numerical evaluation; maximum achieved at δ∗≈3.92888\delta^{\ast} \approx 3.92888

Known lower bounds

Bound Reference Comments
β+ππ≈1.82283\frac{\beta + \pi}{\pi} \approx 1.82283 [BDZ2013]

Here β=π(2γ+2log⁡2+3log⁡π−4log⁡Γ(1/4))\beta = \pi(2\gamma + 2\log 2 + 3\log\pi - 4\log\Gamma(1/4)), where γ\gamma is the Euler–Mascheroni constant.

References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Brezis–Gallouet–Wainger remainder constant on the 2D torus ($C_{16}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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